Comminution or breakage of particles in apparatus such as mills and crushers is an important mineral processing operation. These operations receive run of mine ore from the mine and then reduce the size of the particles before the ore is subjected to further processing to liberate the mineral values within the ore. The capital cost of installed mills and crushers is extremely high. Further the energy demand of mills and crushers is also very high and the efficiency of converting input energy into breakage of particles is extremely low. Consequently there is an ongoing need to develop an improved understanding of these processes so as to enable engineers to operate these processes more efficiently.
The modeling of mineral processing operations is a tool that is widely used throughout the mineral processing operations for this purpose. These models assist in understanding the process and can be used in process development, optimizing plant performance and also in the design of new plants. In particular the model can be used by engineers to help understand the breakage of particles in a mill for example. This understanding in turn can lead engineers to adjust mill settings and this can result in more effective operation in the mill.
The model can also be used for the simulation of plant performance. The simulation of comminution apparatus such as autogenous (AG) and semi-autogenous (SAG) mills, ball mills and crushers is used widely by the mining industry for the design and optimization of plants and also general trouble shooting. However it will be readily appreciated that the value of any simulation model will depend on the underlying accuracy and validity of the models used to define particle breakage in the mill or crusher. Applicant provides simulation services in relation to various comminution apparatus and is aware of limitations in many of the breakage functions used in its simulations. Applicant therefore recognizes the benefits to be obtained from any improvement in the modeling of particle breakage due to impacts of the type that occur in comminution apparatus.
In the modeling of a comminution process the likely size distribution of the progeny particles produced as a result of an impact needs to be determined. Further the effect of feed particle size on the size distribution of the product particles and the influence of the energy applied to the particle by the impact needs to be understood.
The first step in this process is to obtain experimental data on the distribution of product particles as a function of certain impact energies and feed particle size. This is done by conducting particle breakage tests of the type known in the prior art as the pendulum test and the drop test.
These tests involve subjecting a certain size of particle to an impact at a specific energy and then measuring a breakage index that can be converted into the size distribution of the resultant particles. These tests produce a product particle size distribution for the breakage of individual particles as a function of their size and also the amount of applied specific energy that produces the breakage.
A particle size distribution (PSD) can be represented by means of a graph showing the relative amounts of the various sizes on the y axis plotted against the different particle sizes on the x axis. The particle size distribution of a population of broken or product particles produced as a result of a collision or impact can be represented by means of a particle size distribution graph, e.g. a cumulative size distribution that plots the cumulative percentage by weight of broken particles in a population underneath a certain size against that particle size. The percentage by weight of particles below a given size is determined by classification. FIG. 1 shows a graph of the cumulative size distribution of broken particles produced by an impact.
While it is possible to describe the size distribution of broken particles as a result of a collision by a PSD curve such as that shown in FIG. 1 other ways of representing this information have evolved in this art.
One such technique is to represent this information by means of a breakage index called the t10 index or product fineness index. In essence the t10 index provides all the information necessary to obtain the full PSD. The t10 index can be converted into a full PSD curve by means of a graph known as the family of t-curves which was first published by Narayanan. FIG. 2 shows a graph of the family of t-curves. The form of the t-curves and the manner in which they are derived will be discussed below.
The drop weight tests cause an impact of specific energy to be applied to a particle and the broken particles resulting from the impact are collected and then classified. The broken particles are classified using a standard set of sieves of different sizes. The discrete weight percentages passing through each sieve are then converted to cumulative weight percent of the feed particles passing through the screen.
The weight of particles passing through screens having the sizes of 1/50th, 1/25th, 1/10th, ¼ and ½ of the initial screen size is then calculated from the cumulative PSD by a cubic spline calculation that is known in the art. The cumulative weight percentage passing through the 1/75th screen is called the t75 index, the amount passing the 1/50th screen is called the t50 index, the amount passing the 1/10th screen is called the t10 index and so on. By repeating the process across different energy levels a set of tn indices can be built up that provide the data that enable the family of t-curves to be obtained.
In FIG. 2 the t75, t50, t25, t10 etc indices are plotted on the y axis against the t10 index on the x axis. Several independent researchers in the field including Narayanan have confirmed that the mathematical relationships defined by the t-curves does in fact accurately represent the breakage of particles at different energy levels and that this family of curves is basically universal for rock materials.
Thus in the absence of specific data on the family of t-curves for a given material a default set of curves may be used that applies to most materials such as rock. However if the breakage characteristics of the material being used are quite different from the breakage of rock then actual breakage tests results for that particular material may be used to generate a family of t-curves. However for most mineral ores this will not be necessary and the default set of t-curves may be used. Yet further some of the published information on t-curves for certain materials is presented in the form of reference tables.
The t-curves are very useful because once the t10 is determined for a material, e.g. as a result of drop test, then the t-curves can be used to transform or convert the t10 value into a full cumulative particle size distribution as shown in FIG. 1. Put another way the t10 index is a convenient form of shorthand for representing the particle size distribution that can be obtained from the t-curves or reference tables that are available in the literature, or from t-curves derived from actual breakage tests for that material.
The PSD is obtained from the family of t-curves by extending a vertical line up from the measured t10 index on the x axis and intersecting the tn curves to obtain plot points for t2, t4, t25, t50 and t75. These points are then used to plot out the cumulative PSD.
As described above the t10 index is the weight percentage of the population of broken particles that pass through a screen having openings that are 1/10th of the initial mean size of the particle subjected to the impact. It is a measure of the fineness of the population of particles produced by an impact. The higher the value of t10 the greater is the weight of fines in the product population.
Applicant has over time used the t10 index as a measure or finger print of the particle size distribution of the broken particle population. It is a convenient tool because once the t10 index is established then the full cumulative PSD can be obtained from the t-curves. While the specific choice of t10 to define product fineness is arbitrary in some respects, it has become widely used in the field and this is why it is given weight in this specification.
As indicated briefly above, drop weight tests are used to calculate the product fineness index t10 for the particles that are subjected to the impact tests.
Once the t10 has been calculated for the test particles then the test data is fitted to a model to calculate the material specific parameters for that ore. Once these parameters have been determined they are plugged into the model. The model can only be used to predict the size distribution of broken particles after these parameters have been calculated.
A model that has been used in the prior art to predict the distribution of broken particles as a result of a particle undergoing an impact is the following equation (hereinafter referred to as the prior art JK model).t10=A(1−e−b·Ecs)  (1)                where:        t10 index is the percent by weight of the initial mass of particles passing through a screen having mesh openings that are 1/10th of the initial mean size of the test particles;        Ecs is the specific comminution energy that is applied to the impact expressed in kWh/t;        A and b are impact breakage parameters that depend on the material, e.g. ore that is being broken and these are therefore different for different ores and take into account the different breakage behavior of the different ores.        
This model was developed by the Julius Kruttschnitt Mineral Research Centre (JKMRC) and has been used widely for over twenty years. The equation above for the t10 product fineness is an exponential function. The parameter A is the level at which t10 reaches an asymptote that represents the maximum extent of particle breakage that can be obtained. Additional impact energy above this level does not produce increased levels of particle breakage.
By contrast A*b is the slope of the curve at its initial take away point towards the lower end of the curve. The product of A and b is used for comparison between different materials.
The breakage properties of a particular ore are conveniently characterized and expressed by the product of A*b. Many mining companies have developed extensive databases of A and b parameters and also A*b results for many different ore bodies. The database of A and b values are generally developed over a long period of time and can be used repeatedly by the company. The A*b value is characteristic of the particular rock that has been tested. As long as the rock material remains the same, the values of A and b do not need to be re-determined each time the particle breakage of the material is studied.
As indicated briefly above the physical breakage test results produced by the drop weight apparatus are used to calculate characteristic A and b parameters for each ore that is tested. Standard numerical methods such as statistical curve fitting techniques are used to calculate A and b for each particulate material from the drop weight test results. The curve fitting techniques start with estimated values for A and b and then a new t10 is calculated with these parameters. The calculated value of t10 is then compared with the experimentally determined value of t10 from the drop weight tests to calculate the error. This error is then divided by the standard deviation and then squared. This process of calculating a new t10 with the estimated parameters A and b is then repeated for all different particle sizes and each energy level of each particle size. This produces a squared error for each of the test results. The squared errors are then summed to yield a sum of the squared errors. Based on this result new estimates are selected for A and b and another iteration of the same sequence of calculations is carried out to calculate a new set of t10 values. This produces another sum of the squared errors that can be compared with the previous sums of the squared errors. These iterations are repeated until a minimum sum of the squared errors is obtained which fixes the values of A and b.
The values of parameters A and b that produce the minimum sum of the squared errors represent the best fit of the model to the experimental data and these are the A and b values calculated for that material.
One limitation of using the prior art JK model defined in Equation 1 is that the model calculates one value for parameter A and one value for parameter b for a given particle sample which has a range of particle sizes. The assumption implicit in this model therefore is that the breakage parameters are not affected by particle size. Put another way it assumes that particles of different sizes would be broken in the same way when subjected to the same impact energy. Thus the model effectively calculates an average set of A and b parameters for all particle sizes.
Applicant has shown some test work with a Mt Coot-tha quarry material having average particle sizes covering at least the range of 10.3 mm to 57.8 mm.
The particles of the quarry material were subjected to testing with a drop weight tester to determine the t10 index for six different particle sizes using a number of different energies for each particle size. FIG. 3 shows the drop weight test results for the Mt Coot-tha material as well as the curve fitting calculations used to fit the JK model to the data to obtain values for the parameters A and b. In addition the calculations that were carried out to calculate A and b by the numerical curve fitting techniques described above are also shown in FIG. 3. This results in a value for the parameter A of 59.07 and parameter b of 0.435 for this Mt Coot-tha quarry material.
The JK prior art model was then plotted on a graph as a single curve of t10 against specific breakage energy (Ecs) and this graph is shown in FIG. 4 of the drawings. The individual points obtained from the drop weight test data were also plotted on this curve. This graph therefore shows how closely the model fits the test results. It is clear from the graph in FIG. 4 that the single curve representing the model reflects an average curve for the test results across the different test particle sizes. There are data points above the model line and data points below the line.
Applicant recognized the limitations of the prior art JK model illustrated by the graph in FIG. 4 and started to investigate the influence of particle size on particle breakage.
In order to demonstrate the influence of particle size on particle breakage Applicant calculated the model parameters A and b separately for each of the different sizes of particles of the Mt Coot-tha material in FIG. 3. The different values for A and b determined for each particle size are shown in the table of FIG. 5. The differences in the values of A and b calculated for the different particle sizes were significant for some particle sizes.
Each of the parameters A and b was then used to plot individual curves for each of the particle sizes using the JK model defined in Equation 1. FIG. 6 shows a family of curves with each curve representing the JK prior art model applied to a different particle size. In essence this family of curves shows that different sizes of particles have different A and b values and that the material breakage parameters are in fact a function of the size of the particle as well as the material that is being broken.
As a general proposition the JK model curves of the larger particles sit higher than the curves for the smaller particles. This tends to indicate that larger particles are easier to break than smaller particles. This accords with the Applicant's experimental work and Applicant believes that this can be explained by the fact that the crack density of larger particles is much greater than that of smaller particles.
Further each of the individual curves calculated for one particle size represents a close and consistent fit with the test data showing that it fitted the data better than the FIG. 4 graph. This shows that the underlying assumption in the prior art JK model that different sized particles break in the same way is not accurate.
In a particle breakage apparatus such as an AG/SAG mill the feed stream of particulate material typically contains a wide range of particle sizes. For example the feed stream may comprise particles having a size range from 200 mm down to smaller than 1 mm. In a crusher the range of feed particle sizes can be even greater than this. Clearly therefore it would be advantageous if a method of characterizing the breakage of an ore could be devised that took into account the effect of size on the breakage of the particles. This would lead to an improved method of characterising ore breakage. This in turn would have the potential to lead to improved modeling of particle breakage in comminution apparatus, in particular discrete element modeling.
Applicant has looked at ways of developing the JK model to account for the effect of particle size. If the effect of particle size could be worked into the model it would have the potential to produce an improved fit with the test results.
Applicant initially looked at adapting the prior art JK model to account for different particle sizes. For example, Applicant experimented with ways of incorporating the effects of particle size into the parameters A and b used in the prior art model Equation 1. Specifically, Applicant investigated whether there is a relationship between initial particle size and the product of A and b calculated for the different sizes of particles.
In FIG. 7 A*b has been plotted against the initial mean particle size for each particle size. As is clearly shown in the graph no consistent relationship between the fitted values of A and b for each size and initial particle size could be established. The shape of the graph was quite different for different particles. Further the individual plots of A*b versus size did not indicate any relationship between A*b and particle size that could be described by a mathematical formula.
It would have been useful if such a relationship had existed between A*b and particle size as the prior art JK model could have then been adapted by defining the relationship between A*b and particle size and inserting the relationship into the prior art model equation. However, no mathematical relationship between the parameters A and b and particle size could be defined.
Accordingly, there is a need to develop a new model for determining breakage properties, and more particularly the likely distribution of broken particles, when a feed particle of a certain material, for example an ore, is subjected to an impact with a certain amount of energy wherein the model takes into account the effect of the size of the feed particle.